Phases in the gaugino sector: direct reconstruction of the basic parameters and impact on the neutralino pair production

We consider recovering analytically the (generally complex) parameters $\mu$, $M_1$ and $M_2$ of the gaugino and Higgsino Lagrangian, from appropriate physical input in the chargino and neutralino sectors. For given $\tan\beta$, we obtain very simple analytic solutions for $M_2$, $| \mu|$, $Arg[\mu]$ in the chargino sector and a twofold $| M_1 |$, $Arg[M_1]$ analytic solution in the neutralino sector, assuming two chargino, two neutralino masses, and one of the chargino mixing angles as physical input. The twofold ambiguity in the neutralino parameters reconstruction may be essentially resolved by measuring the $e^+e^- \to \chi^0_1 \chi^0_2$ production cross-section at future linear collider energies, which we study explicitly with the phase dependences. Some salient features and specific properties of this complex case gaugino "spectrum inversion" are illustrated and compared with the similar inversion in the real case. In particular, our algorithms exhibit in a direct and transparent way the non-trivial theoretical correlation among the chargino and neutralino parameters, and the resulting allowed domains when only a subset of the required physical input masses and production cross-sections is known.